A 14.55-year maturity zero-coupon bond selling at a yield to maturity of 7% (eff
ID: 2762021 • Letter: A
Question
A 14.55-year maturity zero-coupon bond selling at a yield to maturity of 7% (effective annual yield) has convexity of 160.0 and modified duration of 13.45 years. A 40-year maturity 5% coupon bond making annual coupon payments also selling at a yield to maturity of 7% has nearly identical modified duration—-13.25 years—-but considerably higher convexity of 280.0. a. Suppose the yield to maturity on both bonds increases to 8%. What will be the actual percentage capital loss on each bond? What percentage capital loss would be predicted by the duration-with-convexity rule? (Do not round intermediate calculations. Round your answers to 2 decimal places.) Zero-Coupon Bond Coupon Bond Actual loss % % Predicted loss % % b. Suppose the yield to maturity on both bonds decreases to 6%. What will be the actual percentage capital gain on each bond? What percentage capital gain would be predicted by the duration-with-convexity rule? (Do not round intermediate calculations. Round your answers to 2 decimal places.) Zero-Coupon Bond Coupon Bond Actual gain % % Predicted gain % %
Explanation / Answer
Present value of annuity factor (PVAF)= {1 – (1+r)-n}/r
PVAF at 6% for 40 years = (1 – 1.06-40)/0.06 = 15.0463
PVAF at 7% for 40 years = (1 – 1.07-40)/0.07 = 13.3317
PVAF at 8% for 40 years = (1 – 1.08-40)/0.08 = 11.9246
Present value factor (PVF) = 1 / (1+r)n
PVF at 6% for 40 years = 1/1.0640 = 0.0972
PVF at 7% for 40 years = 1/1.0740 = 0.0668
PVF at 8% for 40 years = 1/1.0840 = 0.0460
PVF at 6% for 14.55 years = 1/1.0614.55 = 0.4284
PVF at 7% for 14.55 years = 1/1.0714.55 = 0.3737
PVF at 8% for 14.55 years = 1/1.0814.55 = 0.3264
Price of bond = present value of annuity of coupon payments + present value of face value
ZERO COUPON BOND
5% COUPON BOND
YTM = 6%
YTM = 7%
YTM = 8%
YTM = 6%
YTM = 7%
YTM = 8%
Face value
$1,000.00
$1,000.00
$1,000.00
$1,000.00
$1,000.00
$1,000.00
Coupon rate
5.00%
5.00%
5.00%
Annual coupon payment
$50.00
$50.00
$50.00
PVAF
15.0463
13.3317
11.9246
Present value of coupon payments
$752.32
$666.59
$596.23
PVF
0.4284
0.3737
0.3264
0.0972
0.0668
0.046
Present value of face value
$428.40
$373.70
$326.40
$97.20
$66.80
$46.00
Price of bond
$428.40
$373.70
$326.40
$849.52
$733.39
$642.23
A. If YTM increases to 8%
Zero Coupon bond
Actual capital loss = ($326.40 - $373.70) / $373.70 = 12.66%
Predicted percentage capital loss in duration with convexity rule = {(-Duration/1+r)*Change in r} + {0.05 * Convexity * (Change in r)2}
Duration with convexity capital loss = [(-13.45/1.07)*(0.01)] + [0.5 * 160 * (0.01)2] = -0.1177 = 11.77%
5% Coupon Bond
Actual capital loss = ($642.23 - $733.39)/$733.39 = 12.43%
Duration with convexity capital loss = [(-13.25/1.07)*(0.01)] + [0.5 * 280 * (0.01)2] = -0.1098 = 10.98%
B. If YTM increases to 7%
Zero Coupon bond
Actual capital gain = ($428.40 - $373.70) / $373.70 = 14.64%
Duration with convexity capital gain = [(-13.45/1.07)*(-0.01)] + [0.5 * 160 * (0.01)2] = -0.1177 = 13.37%
5% Coupon Bond
Actual capital gain = ($849.52 - $733.39)/$733.39 = 15.83%
Duration with convexity capital gain = [(-13.25/1.07)*(-0.01)] + [0.5 * 280 * (0.01)2] = -0.1098 = 13.78%
ZERO COUPON BOND
5% COUPON BOND
YTM = 6%
YTM = 7%
YTM = 8%
YTM = 6%
YTM = 7%
YTM = 8%
Face value
$1,000.00
$1,000.00
$1,000.00
$1,000.00
$1,000.00
$1,000.00
Coupon rate
5.00%
5.00%
5.00%
Annual coupon payment
$50.00
$50.00
$50.00
PVAF
15.0463
13.3317
11.9246
Present value of coupon payments
$752.32
$666.59
$596.23
PVF
0.4284
0.3737
0.3264
0.0972
0.0668
0.046
Present value of face value
$428.40
$373.70
$326.40
$97.20
$66.80
$46.00
Price of bond
$428.40
$373.70
$326.40
$849.52
$733.39
$642.23
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